Nuclear Instruments and Methods in Physics Research A 375 (1996) 441-444 NUCLEAR INSTRUMENTS

a METHODS IN PMVSICS RESEARCH

Sectlon A ELSEVIER

Dependence of the VUV-FEL performance at the TESLA Test Facility on magnetic field errors

B. Faatza.“, J. Pfliiger”, P Pierinib

‘Hamburger S~ncllrotronstrallrnRslabor, HASYLAB at Deutsches Elektron S.vnchrotron, DESY, Notkestr. 85, 2260.3 Hamburg, Germany ‘INFN Milano-LASA, Via Cervi. 2001, 20090 Segrate (MI), Italy

Abstract The VUV FEL, which is in its design stage at DESY, is a SASE device which is driven by a I GeV superconducting linac.

In this paper, the performance of the 25 m long undulator, is studied. Results of simulations, including magnetic peak-field errors, are presented. Dependence of the performance on these errors is discussed.

1. Introduction

In designing a VUV-FEL, much effort is spent in

reducing the electron beam energy spread and emittance. As one aims at shorter wavelengths, the phase space

volume of the electron beam has to become smaller. For a

SASE FEL, the undulator needs about 1000 periods. In this case the cumulative effect of magnetic field errors becomes another issue to take into account. During the past decade, both analytical and numerical studies have shown that two important parameters which determine the FEL perform ante are the phase of the electron with respect to the ponderomotive potential, as well as the beam wander, i.e.. the deviation of the electron beam from the undulator axis [l-7]. The first quantity determines the energy exchange

between electrons and the field, the second determines the transverse overlap between electron beam as a whole and the radiation field.

The different error sources, such as variation in un- dulator wavenumber or random tilt of the poles, generating small undesirable magnetic field contributions in both transverse and longitudinal directions, are assumed to be either negligibly small, or are assumed to be well described by an effective random variation of the main peak field of the planar structure. The aim of this paper is to determine the influence of phase and overlap for the TTF-FEL parameters on the gain as well as to establish some general idea within what distance the electron beam has to remain from the optical axis, including the motion due to the FODO-lattice used to keep the beam confined within the specified radius of 57 pm. The design of the undulator

*Corresponding author. Tel. +49 40 8998 2694. fax +49 40 8998 2787, e-mail faatz@vxdesy.desy.de.

with a superimposed FODO-lattice is studied in separate

papers [S-9]. This paper is organized as follows. In Section 2, some

comments will be made on the influence of the error in

magnetic peak held. The results of simulations for differ- ent error distributions are shown in Section 3 and the

results are discussed and first estimations on tolerances are given in Section 4.

2. Influence of magnetic field errors

From FEL physics, it is well known that two important quantities determine to large extent the FEL performance.

One is the resonance condition, i.e., the phase of the electrons with respect to the ponderomotive potential, the second is the overlap between electron beam and field. The

ponderomotive phase is approximated by

with

A@(z) = k,

where ku and k, are the undulator and radiation wavenumber, respectively, ‘yO and p, are the energy and transverse electron velocity normalized to the speed of light. Using (4’) = 0 to determine the resonant wavelength results in the following resonance condition

A.=~(l+~)

0168.9002/96/$15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved

SSDI 0168-9002(95)01 199-4 VII. FEL TECHNOLOGY

442 B. Faatz et al. / Nucl. Instr. and Meth. in Phys. Rex. A 375 (1996) 441-444

With the resonant wavelength given by Eq. (2), we are now ready to calculate the phase shake. Its rms value is defined as the phase at the magnetic poles including errors minus the phase in case of ideal magnetic field squared and

summed over all poles (see for example Ref. [5]). With the definition of phase as given by Eq. (I), the ideal Ic, is zero

on the poles. Thus, the rms phase shake is

where ICI, is the phase evaluated on the poles. Poles are labeled by their appropriate number up to 2N,, with N” the number of undulator periods.

In case of a non-ideal undulator, the derivative of p,

depends on the error of the nth pole. Because the integral up to the nth pole does not vanish, & depends on all previous magnetic peak-field errors. Therefore, the electron beam trajectory may substantially deviate from the optical

axis. Similar to the phase-shake, we also define

(4)

where x,, is the second field integral evaluated on the

magnetic poles, i.e., x(z) = J &(z’) dz’. In order to describe a realistic undulator, one has to take

into account the possibility to “tune” the matching sections that set the first and second field integral equal to zero. A cos-field is assumed with constant peak field in

case of an ideal undulator. Matching is achieved by starting and ending with quarter-periods instead of half. In

all calculations it is assumed that the field amplitude of these two poles are adjusted such that first and second field

integrals over the whole undulator field are equal to zero. The FODO-lattice is not taken into account in these adjustments.

3. Results of simulations

In this section, the model is applied to the TTF VUV- FEL (see Table 1 for details). Simulations have been

performed with TDA3D [lo-l 11. Results given in Figs. 1 and 2 are obtained without

FODO-lattice. The emittance is taken equal to zero. All other parameters are as in Table 1. In order to distinguish between phase shake, c~, and electron beam wander, o;, the normalized power (P,,,lP,“) shown in Fig. 1 is obtained by using an error per undulator period instead of an error per half-period. Thus, the field integral per period is always zero and, consequently, the electron beam remains on axis. Of the two phenomena studied in this paper, only phase-shake can become large in this case. In order to get an rms phase-shake of the order of 1 rad for an

Table I Parameters of the TTF VUV-FEL

Electron

Energy

Peak current

Normalized rms emittance

Rms energy spread

External p-function

Rms beam size in

the undulator

Undulator

Type Period

Peak magnetic field

Magnetic gap

Undulator module length

Effective undulator length

FODO-period

Quadrupole strength

Radiation

1 GeV

2.500 A

2n mm mrad

0.1%

3m

57 urn

Planar

27.3 mm

0.497 T

12mm

4.4 m

25 m

1.2 m

4Tfm

6.4 nm

undulator module of 4.4 m, as shown in Fig. 1. the field- amplitude error had to be increased to as much as 4%. As can be seen, up to 0.5 rad, there is only a few percent decrease in power.

In Fig. 2. errors are chosen per half period. Therefore, the electron beam can wander off-axis in this case. In all calculations, the rms field-amplitude error is OS%, which is believed to be feasible. Fig. 2a shows again the

normalized power versus rms phase-shake. As can be seen, the power reduction is much larger than in the case with an error per period. The scatter of the points is much larger

than in Fig. 1. In Fig. 2b, the normalized power versus

45 _

40 -a.** . l l - 0 0.

0. 35 l -

.G 25 a . 20

l

s 0 15

a 10

5L I 01 I , I I 0.0 0.5 1 .o 1.5 2.0

cv (rad)

Fig. I. Normalized power versus rms phase-shake in case of an undulator field error per period after the first undulator module.

Calculations have been performed in the one-dimensional limit.

i.e., without FODO-lattice. Parameters as in Table 1.

B. Faatz et al. I Nucl. Instr. and Meth. in Phy. Res. A 375 (1996) 441-444 443

ri’ .

5

a”

45 0

40 - l - l e

35 r (a)

30 1 l l

25

20

15

IO :

l

l l

l l

l e l

5 l S l

0 I I I I I ,I,,1

0.0 0.2 0.4 0.6 0.8 1 .o 1.2

q+, WI 45 , 1

ti’ 25

’ 20 5

a0 15

‘.

l .+ . .

‘. l .

e 10 - 6..

l ? 5- -i-.. l

0 I I I I I 1 , , “.,a-- .I....( 9 ,

0 20 40 60 80 100 120

~~ Wn)

Fig. 2. Normalized power versus rms phase-shake (a) and versus

electron beam wander (b) in case of an undulator field error per

half-period after the tirst undulator module. All remaining parame-

ters as in Fig. 1

beam wander is shown. The dotted line is a Gaussian fit. As can be seen, the phase-shake is not the principle parameter which determines the FEL performance in this

case. It is dominated by the wander of the beam (see for example Ref. [4]). In calculations so far, the FODO-lattice was not taken into account. In case of an ideal undulator, i.e., without any errors, now including a FODO-lattice. the influence of electron beam mismatch is studied first. By giving the beam an initial transverse offset, it performs an oscillation throughout the whole undulator, thus reducing the overlap between electron beam and radiation field.

In Fig. 3. the normalized power at the exit of one

undulator module is shown for different values of the initial offset. Note that the normalized power in this case does not exceed 21, compared to 42 in the previous figures. This power reduction is caused by the non-zero emittance and focussing of the electron beam. The figure shows that the power reduction due to an initial offset can be approximated by exp(-Xi/2crz), where a, is the electron beam radius.

In Fig. 4, the performance of the FEL is shown

ci’ .

s a”

25

‘e\ ‘\e,

l

0 20 40 60 80 100 120 140

X0 (v-N Fig. 3. Normalized power versus initial beam offset. No undulator

errors are included. Period of the FODO-lattice is 1.2 m.

including magnetic field errors and the FODO-lattice. It has been assumed that the focusing structure is ideal. Errors in this structure will be studied in a future paper. As

can be seen, the results are similar to those obtained without focusing. The case without errors gives a normal-

ized power of approximately 21. as in Fig. 3. With a a, of 10 p,rn, the reduction is 10%. We would like to stress that the value for Us has been calculated with Eq. (4), using the on-axis magnetic field, which is equivalent to neglecting the FODO-lattice. It does give, however, the deviation with respect to the ideal undulator.

In the TTF-FEL, we aim for four steering stations per module. This should result in an overlap which is large

enough not to have a large gain reduction. According to Ref. [4] assuming a gain length of approximately I .4 m (or 50 periods), a peak field error of 0.5% and steering stations

every 1.1 m, we find W = 0.08, and a gain reduction of less than 2% (see Eq. (7.32) in Ref. [4]).

25h 20

15

10

5

0 0 20 40 60 00 100 120

0, km)

Fig. 4. Normalized power versus electron beam wander in case of an undulator field error per half-period after the first undulator

module including the FODO-lattice with a period of 1.2 m. All

remaining parameters as in Fig. 1.

VII. FEL TECHNOLOGY

444 B. Faatz et al. / Nucl. Instr. and Meth. ill Phxs. Rrs. A 27S (1996) &-JJJ

4. Discussion and conclusions

In case of the TTF-FEL, the dominant effect of magnetic

field errors is the wander of the electron beam. Phase- shake with a perfectly aligned electron beam only gives a

minor reduction in gain by a few percent per undulator

module, whereas beam wander for the same rms-field error can give a gain reduction by as much as an order of magnitude, if no additional steering is provided within the

module. Calculations including the FODO-lattice show that

oscillations may not exceed IO pm of maximum excursion (about 0.2q). giving a gain reduction of 10%. With the 4 planned steering stations per module, the gain reduction is

no more than a few percent. We would like to stress that in the results presented in

this paper, only the first undulator section has been taken

into account. If the electron beam is perfectly matched at any section entrance, these results also hold for the five

remaining ones, because the errors in the different sections are independent. The main difference between the first and

all remaining sections is, however. that the beam is not prebunched in these calculations. In order to calculate all remaining sections, we intend to modify TDA3D such that it can describe the modulator setup of the TTF-FEL. To our understanding, however, the first undulator section will be the most crucial one, because the radiation field has to

start up from noise.

Acknowledgement

M.V. Yurkov, for their interest in our work and the discussions we have had.

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We would like to thank Yu.M. Nikitina, H.D. Nuhn, C. Pellegrini, J. Rollbach, E.L. Saldin, E.A. Schneidmiller and